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C₂²×C₇×Dic₃ #66723

order	336
isoclinism	isoclinic class of D₃
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₇ · Schur multiplier (H₂) = C₂³

Aliases

C₂²×C₇×Dic₃

Splittings

derived, Sylow	C₃ ⋊ C₂²×C₄×C₇
Hall	C₃×C₇ ⋊ C₂²×C₄

Subgroups

	subgroup	G/H
center, hypercenter	C₂³×C₇	D₃
commutator, hypocenter	C₃	C₂²×C₄×C₇
Frattini	C₂	C₂²×D₃×C₇
Fitting, socle, prime radical	C₂³×C₃×C₇	C₂
cosocle	C₂×C₃	C₂³×C₇
LCS	G ▷ C₃
UCS	C₁ ◁ C₂³×C₇

108 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 336

A minimal faithful permutation representation — 3 generators on 18 points:

(1,2)(3,4)(6,7)(8,9,10,11)(12,13,14,15,16,17,18);(3,4)(5,6,7);(1,2)

Conjugacy classes & character table

168 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#10620529, C₇×C₃⋊#95, C₂×Dic₃×Dic₇, …
maximal normal subgroup of	#10620529, C₇×C₃⋊#95, C₂×Dic₃×Dic₇, …
maximal quotient of	C₂²×C₇×C₃⋊C₈, C₂×C₇×#208, C₂×C₄×C₇×Dic₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 192);